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An extreme value theory approach for the early
detection of time clusters. A simulation-based
assessment and an illustration to the surveillance of
Salmonella
Armelle Guillou, Marie Kratz, Yann Le Strat
To cite this version:
Armelle Guillou, Marie Kratz, Yann Le Strat. An extreme value theory approach for the early de-tection of time clusters. A simulation-based assessment and an illustration to the surveillance of Salmonella. Statistics in Medicine, Wiley-Blackwell, 2014, 33 (28), �10.1002/sim.6275�. �hal-01311727�
An Extreme Value Theory approach for the early
detection of time clusters. A simulation-based
assessment and an illustration to the surveillance of
Salmonella
A. Guillou
∗, M. Kratz
†, Y. Le Strat
‡Abstract
We propose a new method which could be part of a warning system for the early detection of time clusters applied to public health surveillance data. This method is based on the extreme value theory (EVT). To any new count of a particular infection reported to a surveillance system, we associate a return period which corresponds to the time that we expect to be able to see again such a level. If such a level is reached, an alarm is generated. Although standard EVT is only defined in the context of continuous observations, our approach allows to handle the case of discrete observations occurring in the public health surveil-lance framework. Moreover it applies without any assumption on the underlying unknown distribution function. The performance of our method is assessed on an extensive simulation study and is illustrated on real data from Salmonella surveillance in France.
∗Univ. de Strasbourg & CNRS, IRMA UMR 7501, France; Email: armelle.guillou@math.unistra.fr †ESSEC Business School, CREAR risk research center, France; Email: kratz@essec.edu. M.Kratz is also member of MAP5, UMR 8145, Univ. Paris Descartes, France.
‡French Institute for Public Health Surveillance, Saint-Maurice, France; Email: y.lestrat@invs.sante.fr
1
Introduction
Since the pioneering work of Serfling ([1]), several statistical models have been proposed
to detect time clusters from surveillance data recorded as time series count in a given
geographic area. A time cluster is defined as a time interval in which the observed
number of events is significantly higher than the expected number of events. The term
‘event’ is generic enough to include any event of interest such as a case of illness, an
admission to an emergency department, a death or any other health event.
The early prospective detection of time clusters represents a statistical challenge as the
models must take the main features of the data into account such as secular trends,
seasonality, past outbreaks but are also faced with idiosyncrasies in reporting, such
as delays, incomplete or inaccurate reporting or other artefacts of the surveillance
systems. Reporting delays are particularly problematic for surveillance systems that
are not based on electronic reporting. Concerning syndromic surveillance systems,
the same difficulties are encountered, excepted for the reporting delays because these
surveillance systems are mostly based on electronic reporting.
The main statistical approaches introduced in the literature have been recently
re-viewed by Unkel et al [2] with a discussion of statistical issues involved in
evaluat-ing and comparevaluat-ing methods. As mentioned by Unkel et al [2], the published models
are sometimes presented into broad classes such as regression techniques, time series
methodology or statistical process control ([3]; [4]; [5]) but this presentation is
some-what artificial as several methods can be classified into more than one of these classes.
However, in most cases they are based on two steps: (i) the calculation of an expected
0This work was partially supported by the ESSEC Research Center, project #043.203.5.1.0.7.07.
2000 AMSC Primary: 62G32 ; Secondary: 62G05, 62G30
Keywords: Extreme value theory; Return level; Return period; Outbreak detection; Salmonella; Surveillance.
value of the event of interest for the current time unit (generally a week or a day); (ii)
a statistical comparison between this expected value and the observed value. A
statis-tical alarm is triggered if the observed value is noticeably different from the expected
value.
The first step is based on the past counts, or more often on a sample of the past counts,
that takes the seasonality pattern(s) into account. Thus, the current count is compared
to counts that occurred in the past during the same time periods, e.g. 3 weeks around
the current week from 5 previous years. Alternatively, sinusoidal seasonal components
can be incorporated into regression models to deal with the seasonality and to easily
take secular trends into account. More rarely, models try to reduce the influence of
weeks coinciding with past outbreaks. One solution to avoid that such outbreaks reduce
the sensitivity of the model is to associate low weights to these weeks in the estimation
of the expectation (e.g. [6]).
The intentional release of anthrax in the USA in October 2001 emphasized the need to
develop new early warning surveillance systems ([7]; [8]). These surveillance systems
treat an increasing number of data provided from multiple sources of information ([9]).
One logical consequence is to perform statistical analyses with a daily frequency.
Developing automated statistical prospective methods for the early detection of time
clusters is thus essential. It is important for a public health surveillance agency to run
several statistical methods concomitantly in order to compare the alarms generated by
these methods.
To this aim, we propose in this paper a new approach based on Extreme Value Theory
(EVT) (e.g. [10]) for the early detection of time clusters. EVT has not been used so
far in the context of surveillance; adapting and combining in a specific way EVT tools
introduces an innovative and interesting alternative to existing methods for surveillance
data, not so numerous. One of the main advantages of the EVT approach is that we
underlying distribution.
The main idea of our method is the following. First, we associate to each observation
of the dataset a return period, which corresponds to the time that we expect to be able
to see again such a level (and not earlier). Then, whenever we have a new observation,
we look backward on its return period to see if already at least one observation exceeds
it. If it does, we consider the new observation as ”abnormal” (e.g. the potential start
of an epidemic), in the sense of ”not expected” as defined by our rule, and we set off
an alarm.
To evaluate the method, we ran it on simulated data generated by an extensive
sim-ulation study based on recent work [11]. Then to illustrate the method, we applied
it to the detection of time clusters from weekly counts of Salmonella isolates reported
to the national surveillance system in France. Salmonellosis is a major cause of
bac-terial enteric illness in both humans and animals, with bacteria called Salmonella. In
France, Salmonella is the first cause of laboratory confirmed bacterial gastroenteritis,
of hospitalization and of death. In 2005, a study estimated that between 92 and 535
deaths attributable to non typhoidal Salmonella occurred each year ([12]).
The paper is organized as follows. The surveillance system and the data are presented
in Section 2. A description of our method to check if each new observation corresponds
to an unusual/extremal event is given in Section 3. Applications to simulated data and
real counts of Salmonella are developed in Section 4. A discussion follows in the last
section.
2
Data
We will consider, in our paper, simulated datasets in order to test the performance of
our method and real datasets in order to illustrate it. Concerning the latter, let us
present in this section the real data, the counts of Salmonella, on which we will be
of salmonellosis by performing serotyping of about 10000 clinical isolates each year.
Salmonella surveillance is based on a network of 1500 medical laboratories that
volun-tarily send their isolates. Salmonella enterica serotypes Typhimurium and Enteritidis
represent 70 % of all Salmonella isolates in humans among many hundreds of serotypes;
that is why we consider in this paper mainly these two serotypes. For illustrative
pur-pose, four other less frequent serotypes (Manhattan, Derby, Agona and Virchow) are
also considered; Figure 1 shows the weekly number of isolates for these six serotypes
from January 1, 1995 to December 31, 2008. It highlights the great variability in terms
of seasonality, secular trend and weekly number of reported isolates and frequencies of
unusual events. The number of cases is recorded by week of reporting to the National
Reference Center because some dates of typing are missing.
Figure 1: Weekly counts of isolates reported to the National Reference Center for Salmonella in France, January 1, 1995 to December 31, 2008: (a) Salmonella Man-hattan; (b) Salmonella Derby; (c) Salmonella Agona; (d) Salmonella Virchow; (e) Salmonella Typhimurium; (f) Salmonella Enteritidis.
Let Y = {Yt; t = 1, . . . , T } be the univariate initial time series corresponding to the
number of isolates at time point t for a given serotype. If the series Y exhibits a trend,
we estimate it from a linear regression trend component, then we substract it.
Concerning the possible seasonality of Y , as mentioned by many authors (e.g. [7]; [13]),
seasonal effects may have a strong impact in generating a statistical alarm. A common
way to prepare the dataset is to select counts from comparable periods in past years,
as described in the literature ([6]; [14]). The dataset is restricted to the counts that
occurred during the times within these comparable periods. For instance, if the current
time is t of year y, then only the counts for the n = b(2w + 1) times from t − w to t + w
of years from y − b to y − 1 (b > 1, w > 1) are used. Another way to handle seasonality
is to include seasonal factors in the statistical model running on a restricted dataset,
As an illustration Figure 2 represents the restricted dataset for Salmonella Typhimurium,
for a given current week with w = 3 and b = 5.
Figure 2: Weekly counts of isolates for Salmonella Typhimurium. The current week, represented by the arrow, is the last week of December 2008. The counts that occurred in comparable periods (± 3 weeks) in the five previous years, and used to generate or not an alarm, are represented by the striped bands.
In all the sequel, X = (Xt) will denote the transformed (i.e. the detrended subset) time
series from (Yt). To each new observation will correspond a new sample, indifferently
named X or (Xt), to simplify. Each step of the method will apply on X itself.
3
An EVT approach
Suppose we have at our disposal n successive observations that we consider as
realiza-tions of a sample (Xi) of independent and identically distributed (i.i.d.) non-negative
random variables defined on a probability space (Ω, A, P), from an unknown distribu-tion funcdistribu-tion F . We do not make any assumpdistribu-tion on F and do not need any to develop
our method, which constitutes a great advantage of this approach.
We want to detect extreme events, which justifies an EVT framework. Extreme events
have been extensively studied in the literature (e.g. [10]), with two main approaches
developed in the i.i.d. setting, one based on the distributions of maxima (e.g. [15]) and
the other one based on exceedances above some threshold, known as the
Peak-Over-Threshold (POT) method (e.g. [16]).
Assuming independence might appear as a rough approximation to analyze
epidemio-logical discrete time series; however we obtain quite reasonable and interesting results
under this assumption, usual when introducing an EVT approach. Note also that in
most of the existing methods for surveillance data the residuals from the models are
Moreover, the method we propose also allows handling the case of a discrete
phe-nomenon, whereas the classic EVT method requires continuous distributions only.
We will face this issue of using extreme tools for discrete data in one step of our method,
when having to estimate extreme quantiles (or return levels); instead of estimating
them explicitly as we do in the continuous framework, we will estimate their upper
bounds, using bounds introduced in [17] which do not require any strong assumption
(such as continuity) on the tail behavior of the distribution function. The practitioner,
e.g. a biostatistician or an epidemiologist, is often ready to accept more alarms if he
knows that the upper bound is well estimated and can provide a reasonable ‘worst
case scenario’. In that sense, our method, in particular the use of an upper bound
for the return level, should be viewed as an alternative tool which provides additional
information for the practitioner.
Now let us present the fundamentals of our method.
Associated with a given return period T which corresponds to T time units over the
past, a return level zT is defined as the level expected to be exceeded on average once
every T time units, i.e. such that
E T X i=1 1l{Xi>zT} ! = 1 (1)
where 1l{A} represents the indicator function that is equal to 1 if A is true and to 0
otherwise. The last equality can be rewritten as 1 − F (zT) = 1/T . Hence, the return
level zT corresponds simply to a pT−quantile with pT = 1 − 1/T , zT = F←(1 − 1/T ),
F← denoting the generalized inverse function of F .
Notice that in (1), we may have chosen to replace the right-hand side of the equation by
any small integer, say c. The value of c is related to the height of the threshold above
which observations are considered as exceedances (above this threshold). The smaller
it is preferable to select a high threshold, which means a small value of c. We chose
here to make it really extreme, not asking for more than one exceedance, in average,
above the last observation. We could relax a bit this choice, by taking for instance
c = 2, but we prefer to be on the safe side. As mentioned in the discussion of Section
5, an optimal value of c, or a prescribed value in order to match the performance of
other methods already proposed in the literature is an interesting open question which
will lead to further investigations.
The idea of the method is to associate with each observation xs a return period Ts
defined theoretically as (1 − F (xs))−1 to be able to determine the return period Tt
associated to each new observation xt at time t. Then, we look backwards (and not
forwards as in the ‘standard’ way) in the interval (t − Tt; t) for the existence of an
observation that would exceed xt. If it exists, we set the rule to generate an alarm at
time t based on the fact that, on average, we do not expect two or more exceedances
on (t − Tt; t].
Notice that in our discrete framework it will not be possible to estimate explicitly the
return levels; instead estimated bounds will be considered.
Therefore, after a preliminary analysis of the data and definition of our sample, we will
compute the estimated bounds of the return levels in order to obtain a graph of the
return periods and levels. Then, we will allocate a return period to any new observation
xt to test if t corresponds to a warning time according to our definition.
Looking at extremal events leads us to the crucial problem of high quantile estimation,
well-studied in the EVT literature (e.g. [10]), based on a sample of observations.
However, EVT is only valid in the case where the underlying distribution function F
of these observations is continuous. This is not the case in the surveillance context
where the observations are counts and thus F is discrete. Therefore, we propose to use
instead upper and lower bounds for the return level zt and estimate them as in [17];
any value of t (in particular it holds for large values), it does work for both small and
large samples, and for F continuous or discrete. So it is well-adapted to our context,
when assuming the random variables associated to the i.i.d. observations.
Note that using bounds for a return level zt will imply that the return period defined
theoretically by (1 − F (zt))−1 cannot be explicitly estimated and we have
T` ≤ (1 − F (zt))−1 ≤ Tb (2)
where T` and Tb denote the return periods of the lower and upper bounds `t(u, w, q)
and bt(u, v) respectively, defined in the on-line Web Appendix and estimated below.
Here u, v and w are suitable power functions and q > 1.
Now let us present our method to define an alarm system. For each new observation
xti, i ≥ 1:
• First we check if the new observation is the largest on all the sample. If it is the case, we generate an alarm. This conservative condition may be weakened, applying it only
when observing a local non decreasing trend.
• Otherwise, we proceed as follows.
Step 1: We draw the plot of the return period on the x-axis and the corresponding
estimate of the upper bound of the return level (instead of the return level itself): t, bbt , where ˆ bt= ˆbt b αt, bβt = tˆθn b αt, bβt (1 − 1/t)βbt 1/αbt , with θˆn(α, β) = 1 n n X i=1 xαi,n(i/n)β. (3)
Step 2: We allocate to each observation xti, i ≥ 1, a return time Ti using the previous
deduce the associated return level Ti. Reading an observation as an upper bound of a
return level means that Ti is in fact the lower bound of the theoretical return period
(1 − F (xti))
−1 that should be associated to the observation x
ti, because of (2).
We justify our choice as follows. Considering b`Ti instead of bbTi in the above method
would have led to underestimate the return period associated to the observation xti,
which could be a problem in the context of alarms (it is better to have more alarms
than less), except if the plots t, b`t
and t, bbt
were close enough, but it is generally
not the case for our datasets where b`t appeared approximately as a constant function
of time ([18]).
Step 3: We use the fact that if (Xj) are i.i.d. random variables, then we have for any
time interval I (T ) with length T
E T X i=1 1l{Xi>zT} ! = 1 ⇔ E X i∈I(T ) 1l{Xi>zT} = 1. (4)
This remark is important since we want to define for each new observation a warning
time, which means to look backward in time.
Hence for each new observation xti, to which a return time Ti has been associated (via
Step 2), we will look in the interval (ti − Ti; ti) to see if there exists an observation
exceeding xti, considered as the new exceedances threshold; if it does, we ring an alarm
at this time ti.
• To finish this section, let us summarize our method as an iterative algorithm. For each new observation xti, i ≥ 1:
1. If xti > max j<ti
xj, then generate an alarm time at time ti. Note that this
conser-vative condition may be weakened, applying it only when observing a local non
decreasing trend.
(a) associate the time Ti to xti, read from the plot t, bbt when xti is regarded as a return level bbTi; (b) consider I (Ti) = (ti− Ti, ti];
(c) look for the existence of an observation xt≥ bbTi, for t ∈ (ti− Ti, ti);
if there exists at least such an observation, then generate a warning time at
ti.
Remarks: in practice, if the return period of the new observation xti is larger than the
length of the sample (Xt), then we choose to be conservative by generating an alarm
at time ti. Other choices may of course be considered. Also, in some sense, detecting
outbreaks when two or more ”extreme” observations in a period are observed, instead
of one, might be considered as a ”heuristic rule”. But, as illustrated in the next section,
our detections are in accordance with the ones obtained with other classical approaches
already used in the literature. Nevertheless, a possible alternative, which would require
further investigations, might consist in replacing the expectation by an estimation of
the probability PP
j∈I(Ti)1l{xj≥ˆbTi} > 1
.
• Now to illustrate our method, let us consider the example of the number of Salmonella Virchow isolates. In Figure 3, the x-axis corresponds to the values of t from 2 to 100
weeks and the y-axis to bbt, calculated for the last week of 2008; the two dashed lines
indicate the 95% confidence interval bounds of bt.
Figure 3: The return level/return period graph for Salmonella Virchow, calculated for the last week of 2008. The black curve represents the upper bound of the return level. Dashed curves represent the 95% confidence interval of this upper bound. To the observation y = 20 (respectively y = 15) does correspond on the x-axis bb91(respectively
bb7) from which we deduce the return period equals to T = 91 (respectively T = 7)
4
Applications
4.1
Simulated data
We generated simulated data following the approach proposed by Noufaily et al. [11].
Let us summarize briefly the procedure:
• First, simulated baseline data (i.e. time series of counts in the absence of out-breaks) were generated using a negative binomial model with a mean including
trend and seasonality determined by Fourier terms. More precisely, these data
were simulated from 42 different parameter combinations (called scenarios and
presented in Table 1 in [11]) with different trends, seasonalities, baseline
fre-quencies of counts and dispersions. The simulations of the baseline data use 100
replicates from each scenario of size 624 weeks. The last 49 weeks were used as
current weeks leading to 4900 replicates for each of the 42 scenarios.
• Secondly, outbreaks were simulated both in baseline and current weeks. Given a constant k, the size of a simulated outbreak, starting in week ti, follows a Poisson
distribution with mean equal to k times the standard deviation of the count at
ti. Then outbreak cases were randomly distributed according to a lognormal
distribution with mean 0 and standard deviation 0.5. Finally, four outbreaks
were generated in baseline weeks and one outbreak was generated in current
weeks. Outbreaks start times were chosen randomly. We chose the values of k to
be 3 and 5 in baseline weeks and from 1 to 10 in current weeks.
• Next, we used two measures, named FPR and POD, introduced in [11] to assess the performance of the EVT method. For each scenario, we calculated the false
positive rate (FPR) as the proportion of the current 49 weeks and 100 replicates,
in which the method generated an alarm in the absence of outbreak. The second
calculated. For each scenario and for each current week, if an alarm is generated
at least once between the start and end times of the outbreak, the outbreak is
considered as detected. The POD is the proportion of outbreaks detected in 100
runs.
Figure 4: False positive rates obtained for outbreaks of 3 (a) standard deviations and of 5 (b) standard deviations. Proportions of detected outbreaks of k standard deviations corresponding to the 42 scenarios. Outbreaks of 3 (c) and 5 (d) standard deviations are included in baselines.
Results of this simulation study are presented in Figure 4 showing the FPRs and
the POD when outbreaks of 3 (plots (a), (c)) and 5 (plots (b), (d)) standard
deviations are included in baselines. We show the central estimates of the FPRs
associated to their 95% confidence intervals. As noted in [11], the FPRs are
highest for scenarios 10-12 with low baseline and over-dispersed data. In our
study, the FPR is also high for scenario 16. Concerning the POD, as expected it
increased with k for each scenario. The PODs are not so different when outbreaks
of 3 standard deviations and 5 standard deviations are present in the baselines.
• Finally, we compared the results obtained for each of the 42 simulated scenarios respectively with the EVT approach and with the Model 0 of Farrington’s method
(see Noufaily et al., 2012) run with α = 1%. We chose the Farrington method
for two main reasons. First the aim of the Farrington algorithm was to develop
a robust method for the routine monitoring of weekly reports on infections for
many different pathogens at the Communicable Disease Surveillance Center in
UK. This method was applied in particular to the detection of Salmonella
out-breaks. Second, this method has been applied in France for the surveillance of
human Salmonella since 2006 and for the surveillance of Salmonella isolated in
performances. In 2008, performances of five methods were evaluated for the early
detection of excess legionella cases in France ([20]) concluding that the Farrington
method had the best positive predictive value, equal to 67%. More specifically,
we presented in Figure 5 the POD obtained from the 100 runs for each scenario
with the Farrington method when outbreaks of 3 standard deviations are
in-cluded in baselines. This plot can be compared to Figure 4(c) obtained with the
EVT approach. As expected, this proportion increases with k for each scenario.
For completeness, we also represented on a same plot the false positive rates
obtained respectively with the EVT approach (black circles) and with the
Far-rington method (white triangles). The means represented by the dots together
with the 95% confidence intervals are based again on these 100 replicates. It is
fairly clear that the performances of the methods depend both on the
character-istics of the time series summarized by the different scenarios and the magnitude
of past and current epidemics. However, both methods favor a good specificity
rather than high sensitivity, and they have a close performance, with a slightly
better one for the EVT.
Figure 5: POD obtained with the Farrington method, Model 0 with α = 1%. Propor-tions of detected outbreaks of k standard deviaPropor-tions corresponding to the 42 scenarios. Outbreaks of 3 standard deviations are included in baselines.
Figure 6: False positive rates obtained for outbreaks with the EVT approach (black circles) and with the Farrington method, Model 0 with α = 1%, (white triangles), for 42 scenarios.
4.2
Real data
For each week from January 2000 to December 2008, the EVT method was applied to
two time series presented in Section 2 (Agona and Manhattan). For each time series
Y and for a week t:
1. We note xt the observed number of cases at week t, that we will compare to
selected counts of weeks from t − 3 to t + 3 of years from y − 5 to y − 1 (to handle
the seasonality as explained in Section 2). This restricted dataset corresponds to
the resulting time series X on which we apply our EVT method.
2. We calculate the upper bound for xt regarded as the return level, using equation
(3).
3. We calculate a return period associated to the upper bound of the return level.
In practice we calculate for t = 2 to Tmax (e.g. Tmax = 100) the increasing upper
bounds ˆbt=2, ˆbt=3, . . . , ˆbt=Tmax. The return period is determined by the rank of the
first upper bound greater than the observed number of cases xt.
4. We generate an alarm at time t if:
(a) the return period is larger than the length of the sample X, because we have
not enough past data to claim that xt is not an unusual event;
(b) xtis larger than the maximum number of counts observed in the sample X;
(c) we observe, over the return period, a number of cases greater than xt.
Moreover, in order to reduce the probability that an alarm could be triggered for few
sporadic cases, a standard rule, put as an option in the program because specific to
Salmonella, has been adopted to keep an alarm at week t if at least 5 cases were
ob-served during the 4 last weeks preceding t. This rule has already been applied at the
of the Health Protection Agency) in the UK using the method developed by Farrington
et al. ([6]). It is an empirical rule, which avoids too numerous alarms whenever the
time series concerns a rare serotype only. In this specific case, an alternative could be
to transform equation (1) into EPT
i=11l{Xi>zT}
= c with a suited small value c > 1
as previously discussed.
We applied the EVT method on the weekly number of isolates for serotypes Manhattan
and Agona. We compared the alarms generated by the EVT method with those
gen-erated by the Farrington method (corresponding to Model 0 in [11] with the threshold
value at time t defined as the upper 99% prediction limit).
Figures 7 and 8 represent both the alarms and the weekly counts over time for the
serotypes Manhattan and Agona. Each triangle represents a statistical alarm. Triangles
on the first line represent the alarms generated by the EVT method, whereas the ones
generated by the Farrington method are given on the second line.
Figure 7: Salmonella Manhattan: Weekly counts from January 1, 2000 to December 31, 2008. Roman numerals refer to the quarters of the years. Alarms generated by the EVT method are represented by triangles on the first line. Alarms generated by the Farrington method are represented by triangles on the second line. The documented outbreak is delimited by the two dashed lines.
In Figure 7, the alarms generated by the two methods occurred in the same period
that corresponds to a documented outbreak, delimited by the dashed lines, for the
serotype Manhattan ([21]). From August 2005 to February 2006, a community-wide
outbreak of Salmonella Manhattan infections occurred in France. The investigation
incriminated pork products from a slaughterhouse as being the most likely source of
2005) and the geographical (south-eastern France) occurrence of the majority of cases
and the distribution of products from the slaughterhouse.
Figure 8: Salmonella Agona: Weekly counts from January 1, 2000 to December 31, 2008. Roman numerals refer to the quarters of the years. Alarms generated by the EVT method are represented by triangles on the first line. Alarms generated by the Farrington method are represented by triangles on the second line. The documented outbreak is delimited by the two dashed lines.
In Figure 8, alarms for the serotype Agona are distributed from 2000 to 2008. There
is a concordance between the two methods during three periods. The first period,
corresponding to 5 weeks in August and September 2003, was not documented as an
outbreak. The second concordance period, corresponds to 15 consecutive weeks from
the last week in December 2004 to week 15 in April 2005. This second period is more
interesting as it corresponds to the beginning of a large outbreak of infections in infants
linked to the consumption of powdered infant formula ([22]). The outbreak period,
de-limited by the two dashed lines, took place in two stages: the first stage from week 53
in December 2004 to week 10 in March 2005 and the second from week 11 to week 21.
A total of 47 cases less than 12 months age were identified during the first stage and
94 cases less than 12 months age were identified during the second stage. The third
period corresponds to the week 29 in July 2008. It included five cases, two of them
coming from a foodborne disease outbreak involving piglet consumption, and the three
others being probably sporadic cases.
The EVT method was implemented using R version 2.9 ([23]). The R-code is
avail-able on request. The function called algo.farrington, implemented into the R-Package
5
Discussion
We believe that the EVT method meets a number of requirements, listed by Farrington
et al. ([3]), for the outbreak detection algorithms implemented in surveillance systems.
Indeed, this method is able to monitor a large number of time series which became
an absolute necessity in modern computerized surveillance systems. It can deal with a
wide range of events as it is the case for the Salmonella infections with the routinely
analyses of several hundred serotypes per week. It can handle time series with great
numbers of cases (such as Salmonella Enteritidis) or small numbers of cases (such
as Salmonella Manhattan). Seasonality is taken into account by comparing counts
over the same periods of time. Other methods propose a direct way to treat the past
aberrations, for instance by associating low weights to the weeks coinciding with past
outbreaks. There is no such need when using the EVT method since the return period
is not a constant but depends on each observed count; alarms can then be generated
even if past outbreaks exist. If a past outbreak is contained in the return period
inter-val, then an alarm will be triggered at time t if the observation at t has been exceeded
by this outbreak. Finally, the method is implemented in a function using the R
lan-guage, allowing to run it in an automated procedure with minimal user intervention.
This method could be easily included into the R surveillance package and used by
public health surveillance practitioners. Recently, several papers have shown that R is
routinely used for the early detection of outbreaks in Europe ([25]; [19]; [26]; [27]; [28]).
Although the model developed by Farrington et al. ([6]) became a standard
refer-ence method, routinely used in France since many years and incorporated in several
surveillance systems, the EVT method seems also to be a valuable and interesting tool
for the recognition of time clusters. It could be integrated in the family of outbreak
detection algorithms used by the public health surveillance agencies since developing
en-sure timely public health intervention.
Obviously our approach does not pretend answering all the issues and could be
im-proved. This paper must be viewed as an exploratory study and constitutes a first
step for further developments already quite promising. The main two advantages of
our approach is that it does not require any assumption on the underlying (unknown)
distribution function and leads to a good specificity. Unfortunately, as the Farrington
one, it is less sensitive than other methods. That is why it should not be used solely
but combined with other more sensitive methods. Due to its competitiveness against
standard approaches, it should help practitioners in the detection of epidemics.
The overall potential of our approach, theoretical as practical, together with a fuller
comparison to other methods will be the subject of a forthcoming paper. In particular,
an interesting open question is the choice of an optimal value c for the right-hand side
in equation (1). This question comes back to discuss the choice of the threshold. In
this paper we fixed c = 1. As emphasized in the simulation study, this value leads to
competitive results of our approach with respect to the Model 0 of Farrington’s method
in a large panel of situations, in terms of parameter combinations, trends, seasonalities,
baseline frequencies of counts and dispersions, but this choice might be improved by
an optimal selection. This leads to further investigations.
Acknowledgements
The authors are grateful to an Associate Editor and two anonymous reviewers for their
comments and suggestions that led to a marked improvement of the article. They would
like to thank Angela Noufaily and Paddy Farrington for providing them simulated
datasets and a R code used in Section 4.1 and Francois-Xavier Weill for providing us
method using the R language.
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